The Derivatives-on-Line Pages

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Next: The martingale representation theorem Up: Modelling: Continuous state space Previous: Girsanov's theorem and the Contents

The Black-Scholes formula II

The above relationships would lead to the BS formula that we discussed above. Under the risk neutral measure the discounted payoffs of the option would be equal to its price today, rendering

$\displaystyle C=e^{-rT}\mathbf{E}_{Q}\left[ S\left( T\right) -K\right] ^{+}=e^{... ...\left[ M\left( T\right) \left[ S\left( T\right) -K\right] ^{+}\right] \text{.}$

From this point, straightforward calculations would give the BS formula.

The real beauty of this method though is when we make the assumption of $%% \mathscr{F}\left( t\right)$ -measurable stock returns, volatilities and interest rates, which can well be non-constant. Of course since they are $%% \mathscr{F}\left( t\right)$ -measurable they are not allowed to be affected by sources of uncertainty other than $B\left( t\right)$ . In that case we would have the following market model

The stock price process is

$\displaystyle dS\left( t\right) =\mu \left( t\right) S\left( t\right) dt+\sigma \left( t\right) S\left( t\right) dB\left( t\right)$ ;
The interest rate process is given by $\left\{ r\left( t\right) \right\} _{t\geq 0}$ , making the discounting factor [bond] process

$\displaystyle R\left( t\right) =\exp \left\{ -\int_{0}^{t}r\left( s\right) ds\right\}$ ;
The wealth process would therefore be

$\displaystyle dX\left( t\right)$ $\displaystyle =$ $\displaystyle \left[ r\left( t\right) X\left( t\right) +\left[ \mu \left( t\rig... ...] dt+H\left( t\right) S\left( t\right) \sigma \left( t\right) dB\left( t\right)$

$\displaystyle =$ $\displaystyle r\left( t\right) X\left( t\right) dt+H\left( t\right) S\left( t\r... ...\frac{\mu \left( t\right) -r\left( t\right) }{\sigma \left( t\right) }dt\right]$

$\displaystyle =$ $\displaystyle r\left( t\right) X\left( t\right) dt+H\left( t\right) S\left( t\r... ...igma \left( t\right) \left[ dB\left( t\right) +\theta \left( t\right) dt\right]$ ;
The discounted stock price process would be [using Itô's formula for $f\left( t,S\right) =R\left( t\right) S$ ]

$\displaystyle d\left[ R\left( t\right) S\left( t\right) \right]$ $\displaystyle =$ $\displaystyle R\left( t\right) \left[ -r\left( t\right) S\left( t\right) dt+dS\left( t\right) \right]$

$\displaystyle =$ $\displaystyle R\left( t\right) \sigma \left( t\right) S\left( t\right) \left[ dB\left( t\right) +\theta \left( t\right) dt\right]$ ;
The discounted wealth process would be

$\displaystyle d\left[ R\left( t\right) X\left( t\right) \right]$ $\displaystyle =$ $\displaystyle H\left( t\right) d\left[ R\left( t\right) S\left( t\right) \right]$

$\displaystyle =$ $\displaystyle H\left( t\right) R\left( t\right) \sigma \left( t\right) S\left( t\right) \left[ dB\left( t\right) +\theta \left( t\right) dt\right]$ .

We therefore observe that if we define a measure by its Radon-Nikodym derivative with $\theta \left( t\right) =\frac{\mu \left( t\right) -r\left( t\right) }{\sigma \left( t\right) }$ we obtain the risk neutral probability measure under which the discounted stock prices form martingales, and the discounted wealth forms a martingale no matter what the portfolio $\mathbf{H}$ is. It is straightforward now to show that such a measure does not allow arbitrage opportunities [in exactly the same fashion as in the discrete case].

Under the stock price follows the diffusion

$\displaystyle dS\left( t\right) =r\left( t\right) S\left( t\right) dt+\sigma \left( t\right) S\left( t\right) dB^{Q}\left( t\right)$ .

The derivative prices would be computed as

$\displaystyle \begin{tabular}{ll} European Call & $\mathbf{E}_{P}\left[ M\left(... ...R\left( T\right) \max_{0\leq t\leq T}S\left( t\right) \right] $%% \end{tabular}$

Kyriakos 2003-03-17

$\displaystyle dX\left( t\right)$	$\displaystyle =$	$\displaystyle \left[ r\left( t\right) X\left( t\right) +\left[ \mu \left( t\rig... ...] dt+H\left( t\right) S\left( t\right) \sigma \left( t\right) dB\left( t\right)$
	$\displaystyle =$	$\displaystyle r\left( t\right) X\left( t\right) dt+H\left( t\right) S\left( t\r... ...\frac{\mu \left( t\right) -r\left( t\right) }{\sigma \left( t\right) }dt\right]$
	$\displaystyle =$	$\displaystyle r\left( t\right) X\left( t\right) dt+H\left( t\right) S\left( t\r... ...igma \left( t\right) \left[ dB\left( t\right) +\theta \left( t\right) dt\right]$ ;

$\displaystyle d\left[ R\left( t\right) S\left( t\right) \right]$	$\displaystyle =$	$\displaystyle R\left( t\right) \left[ -r\left( t\right) S\left( t\right) dt+dS\left( t\right) \right]$
	$\displaystyle =$	$\displaystyle R\left( t\right) \sigma \left( t\right) S\left( t\right) \left[ dB\left( t\right) +\theta \left( t\right) dt\right]$ ;

$\displaystyle d\left[ R\left( t\right) X\left( t\right) \right]$	$\displaystyle =$	$\displaystyle H\left( t\right) d\left[ R\left( t\right) S\left( t\right) \right]$
	$\displaystyle =$	$\displaystyle H\left( t\right) R\left( t\right) \sigma \left( t\right) S\left( t\right) \left[ dB\left( t\right) +\theta \left( t\right) dt\right]$ .